Mathematical fallacy

In mathematics, certain kinds of mistaken proof are often exhibited, and sometimes collected, as illustrations of a concept of mathematical fallacy. There is a distinction between a simple mistake and a mathematical fallacy in a proof: a mistake in a proof leads to an invalid proof just in the same way, but in the best-known examples of mathematical fallacies, there is some concealment in the presentation of the proof. For example, the reason validity fails may be a division by zero that is hidden by algebraic notation. There is a striking quality of the mathematical fallacy: as typically presented, it leads not only to an absurd result, but does so in a crafty or clever way.[1] Therefore these fallacies, for pedagogic reasons, usually take the form of spurious proofs of obvious contradictions. Although the proofs are flawed, the errors, usually by design, are comparatively subtle, or designed to show that certain steps are conditional, and should not be applied in the cases that are the exceptions to the rules.

The traditional way of presenting a mathematical fallacy is to give an invalid step of deduction mixed in with valid steps, so that the meaning of fallacy is here slightly different from the logical fallacy. The latter applies normally to a form of argument that is not a genuine rule of logic, where the problematic mathematical step is typically a correct rule applied with a tacit wrong assumption. Beyond pedagogy, the resolution of a fallacy can lead to deeper insights into a subject (such as the introduction of Pasch's axiom of Euclidean geometry[2] and the five color theorem of graph theory). Pseudaria, an ancient lost book of false proofs, is attributed to Euclid.[3]

Mathematical fallacies exist in many branches of mathematics. In elementary algebra, typical examples may involve a step where division by zero is performed, where a root is incorrectly extracted or, more generally, where different values of a multiple valued function are equated. Well-known fallacies also exist in elementary Euclidean geometry and calculus.

Examples exist of mathematically correct results derived by incorrect lines of reasoning. Such an argument, however true the conclusion, is mathematically invalid and is commonly known as a howler. Consider for instance the calculation (anomalous cancellation):

Although the conclusion is correct, there is a fallacious, invalid cancellation in the middle step. Bogus proofs, calculations, or derivations constructed to produce a correct result in spite of incorrect logic or operations were termed howlers by Maxwell.[4] Outside the field of mathematics the term "howler" has various meanings, generally less specific.

Many functions do not have a unique inverse. For instance squaring a number gives a unique value, but there are two possible square roots of a positive number. The square root is multivalued. One value can be chosen by convention as the principal value, in the case of the square root the non-negative value is the principal value, but there is no guarantee that the square root function given by this principal value of the square of a number will be equal to the original number, e.g. the square root of the square of −2 is 2.

Calculus as the mathematical study of infinitesimal change and limits can lead to mathematical fallacies if the properties of integrals and differentials are ignored. For instance, a naive use of integration by parts can be used to give a false proof that 0 = 1.[6] Letting and , we may write:

after which the antiderivatives may be cancelled yielding 0 = 1. The problem is that antiderivatives are only defined up to a constant and shifting them by 1 or indeed any number is allowed. The error really comes to light when we introduce arbitrary integration limits a and b.

Since the difference between two values of a constant function vanishes, the same definite integral appears on both sides of the equation.

The error in each of these examples fundamentally lies in the fact that any equation of the form

has two solutions, provided a ≠ 0,

and it is essential to check which of these solutions is relevant to the problem at hand.[10] In the above fallacy, the square root that allowed the second equation to be deduced from the first is valid only when cos x is positive. In particular, when x is set to π, the second equation is rendered invalid.

Another example of this kind of fallacy, where the error is immediately detectable, is the following invalid proof that −2 = 2. Letting x = −2, and then squaring gives

whereupon taking a square root implies

so that x = −2 = 2, which is absurd. Clearly when the square root was extracted, it was the negative root −2, rather than the positive root, that was relevant for the particular solution in the problem.

Alternatively, imaginary roots are obfuscated in the following:

The error here lies in the last equality, where we are ignoring the other fourth roots of 1,[11] which are −1, i and −i (where i is the imaginary unit). Seeing as we have squared our figure and then taken roots, we cannot always assume that all the roots will be correct. So the correct fourth roots are i and −i, which are the imaginary numbers defined to square to −1.

When a number is raised to a complex power, the result is not uniquely defined (see Failure of power and logarithm identities). If this property is not recognized, then errors such as the following can result:

The error here is that the rule of multiplying exponents as when going to the third line does not apply unmodified with complex exponents, even if when putting both sides to the power i only the principal value is chosen. When treated as multivalued functions, both sides produce the same set of values, being {e2πn | n ∈ ℤ}.

Many mathematical fallacies in geometry arise from using in an additive equality involving oriented quantities (such adding vectors along a given line or adding oriented angles in the plane) a valid identity, but which fixes only the absolute value of (one of) these quantities. This quantity is then incorporated into the equation with the wrong orientation, so as to produce an absurd conclusion. This wrong orientation is usually suggested implicitly by supplying an imprecise diagram of the situation, where relative positions of points or lines are chosen in a way that is actually impossible under the hypotheses of the argument, but non-obviously so. Such a fallacy is easy to expose by drawing a precise picture of the situation, in which some relative positions will be different form those in the provided diagram. In order to avoid such fallacies, a correct geometric argument using addition or subtraction of distances or angles should always prove that quantities are being incorporated with their correct orientation.

As a corollary, one can show that all triangles are equilateral, by showing that AB = BC and AC = BC in the same way.

The error in the proof is the assumption in the diagram that the point O is inside the triangle. In fact, O always lies at the circumcircle of the △ABC (except for isosceles and equilateral triangles where AO and OD coincides . Furthermore, it can be shown that, if AB is longer than AC, then R will lie within AB, while Q will lie outside of AC (and vice versa). (Any diagram drawn with sufficiently accurate instruments will verify the above two facts.) Because of this, AB is still AR + RB, but AC is actually AQ − QC; and thus the lengths are not necessarily the same.

There exist several fallacious proofs by induction in which one of the components, basis case or inductive step, is incorrect. Intuituvely, proofs by induction work by arguing that, if a statement is true in one case, it is true in the next case, and hence by repeatedly applying this it can be shown to be true for all cases. This "proof" shows that all horses are the same colour.

Let us say that any group of N horses is all of the same colour.

If we remove a horse from the group, we have a group of N - 1 horses of the same colour. If we add another horse, we have another group of N horses. By our previous assumption, all the horses are of the same colour in this new group, since it is a group of N horses.

Thus we have constructed two groups of N horses all of the same colour, with N - 1 horses in common. Since these two groups have some horses in common, the two groups must be of the same colour as each other.

Therefore combining all the horses used, we have a group of N + 1 horses of the same colour.

Thus if any N horses are all the same colour, any N + 1 horses are the same colour.

This is clearly true for N = 1 (i.e. one horse is a group where all the horses are the same colour). Thus, by induction, N horses are the same colour for any positive integer N. i.e. all horses are the same colour.

The fallacy in this proof arises in line 3. For N = 1, the two groups of horses have N − 1 = 0 horses in common, and thus are not necessarily the same colour as each other, so the group of N + 1 = 2 horses is not necessarily all of the same colour. The implication "Every N horses are of the same color, then N+1 horses are of the same color" works for any N greater than one, but fails to be true when N=1. The basis case is correct, but the induction step has a fundamental flaw.